Displaying similar documents to “Multipliers of Hardy spaces, quadratic integrals and Foiaş-Williams-Peller operators”

The Marcinkiewicz multiplier condition for bilinear operators

Loukas Grafakos, Nigel J. Kalton (2001)

Studia Mathematica

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This article is concerned with the question of whether Marcinkiewicz multipliers on 2 n give rise to bilinear multipliers on ℝⁿ × ℝⁿ. We show that this is not always the case. Moreover, we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy...

Multipliers of the Hardy space H¹ and power bounded operators

Gilles Pisier (2001)

Colloquium Mathematicae

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We study the space of functions φ: ℕ → ℂ such that there is a Hilbert space H, a power bounded operator T in B(H) and vectors ξ, η in H such that φ(n) = ⟨Tⁿξ,η⟩. This implies that the matrix ( φ ( i + j ) ) i , j 0 is a Schur multiplier of B(ℓ₂) or equivalently is in the space (ℓ₁ ⊗̌ ℓ₁)*. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of H¹ which we call “shift-bounded”. We show that there is a φ which...

Multiplier transformations on H p spaces

Daning Chen, Dashan Fan (1998)

Studia Mathematica

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The authors obtain some multiplier theorems on H p spaces analogous to the classical L p multiplier theorems of de Leeuw. The main result is that a multiplier operator ( T f ) ( x ) = λ ( x ) f ̂ ( x ) ( λ C ( n ) ) is bounded on H p ( n ) if and only if the restriction λ ( ε m ) m Λ is an H p ( T n ) bounded multiplier uniformly for ε>0, where Λ is the integer lattice in n .

Spherical summation : a problem of E.M. Stein

Antonio Cordoba, B. Lopez-Melero (1981)

Annales de l'institut Fourier

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Writing ( T R λ f ) ^ ( ξ ) = ( 1 - | ξ | 2 / R 2 ) + λ f ^ ( ξ ) . E. Stein conjectured j | T R j λ f i | 2 1 / 2 p C j | f j | 2 1 / 2 p for λ > 0 , 4 3 p 4 and C = C λ , p . We prove this conjecture. We prove also f ( x ) = lim j T 2 j λ f ( x ) a.e. We only assume 4 3 + 2 λ < p < 4 1 - 2 λ .

Transference and restriction of maximal multiplier operators on Hardy spaces

Zhixin Liu, Shanzhen Lu (1993)

Studia Mathematica

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The aim of this paper is to establish transference and restriction theorems for maximal operators defined by multipliers on the Hardy spaces H p ( n ) and H p ( n ) , 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an L ( n ) function m is a maximal multiplier on H p ( n ) if and only if it is a maximal multiplier on H p ( n ) . As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered. ...

A Marcinkiewicz type multiplier theorem for H¹ spaces on product domains

Michał Wojciechowski (2000)

Studia Mathematica

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It is proved that if m : d satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the H 1 space on the product domain d 1 × . . . × d k . This implies an estimate of the norm N ( m , L p ( d ) of the multiplier transformation of m on L p ( d ) as p→1. Precisely we get N ( m , L p ( d ) ) ( p - 1 ) - k . This bound is the best possible in general.

Calderón-Zygmund operators acting on generalized Carleson measure spaces

Chin-Cheng Lin, Kunchuan Wang (2012)

Studia Mathematica

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We study Calderón-Zygmund operators acting on generalized Carleson measure spaces C M O r α , q and show a necessary and sufficient condition for their boundedness. The spaces C M O r α , q are a generalization of BMO, and can be regarded as the duals of homogeneous Triebel-Lizorkin spaces as well.

Multilinear Fourier multipliers with minimal Sobolev regularity, I

Loukas Grafakos, Hanh Van Nguyen (2016)

Colloquium Mathematicae

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We find optimal conditions on m-linear Fourier multipliers that give rise to bounded operators from products of Hardy spaces H p k , 0 < p k 1 , to Lebesgue spaces L p . These conditions are expressed in terms of L²-based Sobolev spaces with sharp indices within the classes of multipliers we consider. Our results extend those obtained in the linear case (m = 1) by Calderón and Torchinsky (1977) and in the bilinear case (m = 2) by Miyachi and Tomita (2013). We also prove a coordinate-type Hörmander integral...

Fourier coefficients of continuous functions and a class of multipliers

Serguei V. Kislyakov (1988)

Annales de l'institut Fourier

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If x is a bounded function on Z , the multiplier with symbol x (denoted by M x ) is defined by ( M x f ) ^ = x f ^ , f L 2 ( T ) . We give some conditions on x ensuring the “interpolation inequality” M x f L p C f L 1 α M x f L q 1 - α (here 1 &lt; p &lt; q and α = α ( p , q , x ) is between 0 and 1). In most cases considered M x fails to have stronger L 1 -regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets E Z every positive sequence in 2 ( E ) can be majorized by the sequence { | f ^ ( n ) | } n E for some continuous funtion f with spectrum...

ω-Calderón-Zygmund operators

Sijue Wu (1995)

Studia Mathematica

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We prove a T1 theorem and develop a version of Calderón-Zygmund theory for ω-CZO when ω A .

Sharp spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

Peng Chen (2013)

Colloquium Mathematicae

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We consider an abstract non-negative self-adjoint operator L acting on L²(X) which satisfies Davies-Gaffney estimates. Let H L p ( X ) (p > 0) be the Hardy spaces associated to the operator L. We assume that the doubling condition holds for the metric measure space X. We show that a sharp Hörmander-type spectral multiplier theorem on H L p ( X ) follows from restriction-type estimates and Davies-Gaffney estimates. We also establish a sharp result for the boundedness of Bochner-Riesz means on H L p ( X ) . ...

Multipliers for the twisted Laplacian

E. K. Narayanan (2003)

Colloquium Mathematicae

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We study ¹ - L p boundedness of certain multiplier transforms associated to the special Hermite operator.

Two problems of Calderón-Zygmund theory on product-spaces

Jean-Lin Journé (1988)

Annales de l'institut Fourier

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R. Fefferman has shown that, on a product-space with two factors, an operator T bounded on L 2 maps L into BMO of the product if the mean oscillation on a rectangle R of the image of a bounded function supported out of a multiple R’ of R, is dominated by C | R | s | R | - s , for some s &gt; 0 . We show that this result does not extend in general to the case where E has three or more factors but remains true in this case if in addition T is a convolution operator, provided s &gt; s 0 ( E ) . We also show that the Calderon-Coifman...

Fejér means of two-dimensional Fourier transforms on H p ( × )

Ferenc Weisz (1999)

Colloquium Mathematicae

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The two-dimensional classical Hardy spaces H p ( × ) are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from H p ( × ) to L p ( 2 ) (1/2 < p ≤ ∞) and is of weak type ( H 1 ( × ) , L 1 ( 2 ) ) where the Hardy space H 1 ( × ) is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ H 1 ( × ) L l o g L ( 2 ) converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on H p ( × ) whenever 1/2 < p < ∞. Thus, in case f ∈ H p ( × ) , the...